On evolution quasi-variational inequalities and implicit state-dependent sweeping processes

Samir Adly, ,Tahar Haddad

doi:10.3934/dcdss.2020105

Abstract

In this paper, we study a variant of the state-dependent sweeping process with velocity constraint. The constraint \begin{document}$ {C(\cdot, u)} $\end{document} depends upon the unknown state \begin{document}$ u $\end{document} , which causes one of the main difficulties in the mathematical treatment of quasi-variational inequalities. Our aim is to show how a fixed point approach can lead to an existence theorem for this implicit differential inclusion. By using Schauder's fixed point theorem combined with a recent existence and uniqueness theorem in the case where the moving set \begin{document}$ C $\end{document} does not depend explicitly on the state \begin{document}$ u $\end{document} (i.e. \begin{document}$ C: = C(t) $\end{document} ) given in [ 4 ], we prove a new existence result of solutions of the quasi-variational sweeping process in the infinite dimensional Hilbert spaces with a velocity constraint. Contrary to the classical state-dependent sweeping process, no conditions on the size of the Lipschitz constant of the moving set, with respect to the state, is required.

Highlights

Translating the dynamic (1) to a mechanical language, we obtain the following interpretation: - if the position u(t) of a material point lies in the interior of the moving set C(t), u (t) = 0, which means that the material point remains at rest; - when the boundary of the moving set C(t) catches up the material point, this latter is pushed in an inward normal direction by the boundary of C(t) to stay inside the moving set and satisfies the constraint
The aim of the present paper is to show how a fixed point approach can lead to the existence theorem for the implicit differential inclusion described by (2)
We handled the situation using the Schauder fixed point Theorem which requires a compactness assumption of the moving set

Summary

Introduction

Translating the dynamic (1) to a mechanical language, we obtain the following interpretation: - if the position u(t) of a material point lies in the interior of the moving set C(t), u (t) = 0, which means that the material point remains at rest; - when the boundary of the moving set C(t) catches up the material point, this latter is pushed in an inward normal direction by the boundary of C(t) to stay inside the moving set and satisfies the constraint. Moreau’s sweeping process, evolution quasi-variational inequalities, state-dependent sweeping process

Objectives

Conclusion